Gale Diagrams Research Articles

Chemical applications of topology and group theory

Etude systematique des isomerisations non planes de polyedres a 5 et 6 sommets, en utilisant leurs diagrammes de Gale

Quotient polytopes of cyclic polytopes part I: Structure and characterization

We investigate the quotient polytopesC/F, whereC is a cyclic polytope andF is a face ofC. We describe the combinatorial structure of such quotients, and show that under suitable restrictions the pair (C, F) is determined by the combinatorial type ofC/F. We describe alternative constructions of these quotients by “splitting vertices” of lower-dimensional cyclic polytopes. Using Gale diagrams, we show that every simpliciald-polytope withd+3 vertices is isomorphic to a quotient of a cyclic polytope.

Constructions for projectively unique polytopes

A convex polytope P is projectively unique if every polytope combinatorially isomorphic to P is projectively equivalent to P. In this paper are described certain geometric constructions, which are also discussed in terms of Gale diagrams. These constructions are applied to obtain projectively unique polytopes from ones of lower dimension; in particular, they lead to projectively unique polytopes with many vertices.

A construction for projectively unique polytopes

A method for constructing projectively unique polytopes with many vertices is described. The method makes use of the properties of Gale diagrams.

Representations of polytopes and polyhedral sets

In this paper we describe a variant of the diagram techniques, such as Gale diagrams for polytopes and positive diagrams for positive bases, which is appropriate for polyhedral sets. We obtain our new technique as a translation-invariant representation of polytopes or polyhedral sets. This approach leads naturally to simpler proofs of the familiar combinatorial diagram relationships. However, this method is more versatile than those previously employed, in that it can be used to investigate linear systems of polyhedral sets, and metrical properties such as volume. In particular, we give an easy proof of a result of Meyer on decomposability of polytopes, and a more perspicuous way of looking at the well-known theorem of Minkowski on the realizability of polytopes whose facets have given outer normal vectors and areas.

Sections and projections of convex polytopes

Introduction . Gale diagrams, and similar techniques, have been used in several recent investigations into the combinatorial properties of convex polytopes. Here we describe other applications, namely to problems concerning sections and projections of polytopes of a given combinatorial type. For example, in §4 we shall show that every n –dimensional convex polytope with at most n + 2 vertices possesses the universal shadow boundary (usb) property : If is a subcomplex of the boundary complex ℬ ( P ) of P , and set is homeomorphic to an ( n – 2)-sphere, then there exists a polytope P', combinatorially isomorphic to P , such that the complex corresponding to is a shadow boundary of P' . We shall also show that this is the best possible result in that polytopes with n + 3 or more vertices do not, in general, have the usb property. A second application of the method, to be described in §5, is to the construction of non-realisable complexes.

Polyhedral diagrams for sections of the non‐negative orthant

A section of the non-negative orthant by an affine subspace is a polyhedral set. A technique, analogous to that of Gale diagrams, is described which enables one to determine the facial structure of such a polyhedral set.

Spherical complexes and radial projections of polytopes

Using the techniques of Gale diagrams a simple criterion is given for determining when a given spherical complex onSn−1 ⊂En is the radial projection, from the centre ofSn−1, of a convex polytope. Previously a criterion was known only for the casen=2.

On zonotopes

In this paper is described a diagram technique for zonotopes, or vector sums of line segments, which is analogous to that of Gale diagrams for general polytopes, and central diagrams for centrally symmetric polytopes. The use of these new zonal diagrams leads to relationships between zonotopes with n n zones of dimensions d d and n − d n - d , and enables one to enumerate all the combinatorial types of d d -zonotopes with n ≦ d + 2 n \leqq d + 2 zones. The connexion between arrangements of hyperplanes in projective space and zonotopes leads to corresponding new results about arrangements.

Polytopes with an Axis Of Symmetry

During the last few years, Branko Grünbaum, Micha Perles, and others have made extensive use of Gale transforms and Gale diagrams in investigating the properties of convex polytopes. Up to the present, this technique has been applied almost entirely in connection with combinatorial and enumeration problems. In this paper we begin by showing that Gale transforms are also useful in investigating properties of an essentially metrical nature, namely the symmetries of a convex polytope. Our main result here (Theorem (10)) is that, in a manner that will be made precise later, the symmetry group of a polytope can be represented faithfully by the symmetry group of a Gale transform of its vertices. If a d-polytope P ⊂ Ed has an axis of symmetry A (that is, A is a linear subspace of Ed such that the reflection in A is a symmetry of P), then it is called axi-symmetric. Using Gale transforms we are able to determine, in a simple manner, the possible numbers and dimensions of axes of symmetry of axi-symmetric polytopes.